Our purpose in these notes is to present a result for a specific Nakayama Algebra. In essence, it affirms that for any order of simple modules, the cyclic Nakayama Algebras with relations (i.e. ) are not standardly st...Our purpose in these notes is to present a result for a specific Nakayama Algebra. In essence, it affirms that for any order of simple modules, the cyclic Nakayama Algebras with relations (i.e. ) are not standardly stratified or costandardly stratified.展开更多
It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic ind...It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.展开更多
Let A = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram An for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we deter...Let A = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram An for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we determine explicitly all the Hochschild homology and cohomology groups of A based on a detailed description of the Bardzell complex. Moreover, the cyclic homology of A can be calculated in the case that the underlying field is of characteristic zero.展开更多
Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and r...Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.展开更多
Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Cala...Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Calabi-Yau properties are generalized without Koszul assumption.We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.展开更多
The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a re...The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.展开更多
文摘Our purpose in these notes is to present a result for a specific Nakayama Algebra. In essence, it affirms that for any order of simple modules, the cyclic Nakayama Algebras with relations (i.e. ) are not standardly stratified or costandardly stratified.
基金supported by Shandong Provincial Natural Science Foundation of China (Grant No.ZR2011AM005)National Natural Science Foundation of China (Grant No.10931006)Shanghai Municipal Natural Science Foundation (Grant No.12ZR1413200)
文摘It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.
基金National Natural Science Foundation of China (Grant Nos.10426014 and 10501010)
文摘Let A = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram An for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we determine explicitly all the Hochschild homology and cohomology groups of A based on a detailed description of the Bardzell complex. Moreover, the cyclic homology of A can be calculated in the case that the underlying field is of characteristic zero.
文摘Abstract Let n ≥ 3. The complex Lie algebra, which is attached to a unit form xixj and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A~, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.
基金supported by National Natural Science Foundation of China(Grant No.11271319)
文摘Poincaré-Birkhoff-Witt(PBW)deformations of Artin-Schelter regular algebras are skew CalabiYau.We prove that the Nakayama automorphisms of such PBW deformations can be obtained from their homogenizations.Some Calabi-Yau properties are generalized without Koszul assumption.We also show that the Nakayama automorphisms of such PBW deformations control Hopf actions on them.
基金This work was partially supported by the Program for New Century Excellent Talents in University (Grant No.04-0522) the National Natural Science Foundation of China (Grant No.10571153).
文摘The main aim of this paper is to give the invariant properties of representations of algebras under cleft extensions over a semisimple Hopf algebra. Firstly, we explain the concept of the cleft extension and give a relation between the cleft extension and the crossed product which is the approach we depend upon. Then, by making use of them, we prove that over an algebraically closed field k, for a finite dimensional Hopf algebra H which is semisimple as well as its dual H*, the representation type of an algebra is an invariant property under a finite dimensional H-cleft extension . In the other part, we still show that over an arbitrary field k, the Nakayama property of a k-algebra is also an invariant property under an H -cleft extension when the radical of the algebra is H-stable.
